Under-investment in Public Clubs

I don't know if I'm re-inventing the wheel. I don't normally do this sort of micro, or any micro.

Talking about "Public Club Investment goods" will sound oxymoronic to an economist. But the investment goods I'm talking about are a bit of both. They look like club goods from one angle, and look like public goods from a different angle. It's the club itself that is a public good, so let's call it a "Public Club". (I probably need a better name.)

[For non-economists: I am using the words "Public" and "Club" the way economists use them, not in the way you use them. A good is "non-rival" if the marginal cost of an additional person consuming the good is zero; a good is non-excludable if it is impossible to prevent an additional person consuming the good without paying. Both Club goods and Public goods are non-rival; but Club goods are excludable and Public goods are non-excludable. Cable TV is a Club good; broadcast TV is a Public Good (it costs nothing to let an additional person watch an existing show, but the cable company can charge and the broadcast company can't charge). Don't argue and split hairs; that was just an example.]

Take a simple two-period example. In the first period the club decides how much to invest. In the second period the club members enjoy the benefits of that investment. Non-members are excluded from enjoying the benefits. But can non-members be excluded from joining the club at the beginning of the second period?

You can see the problem. If club members suffer net costs in the first period, and enjoy net benefits in the second period, each individual has an incentive to postpone joining the club until the beginning of the second period. The investment never gets made. Even if there are positive net benefits for some members in the first period (because the club has some other benefits), the members will under-invest in the first period if they cannot charge new members an additional premium in the second period.

If capital markets are perfect, and in a world of certainty, this problem can be resolved. In the first period the members borrow the funds to finance the investment, so it costs them nothing. In the second period all members (new and old) have to pay dues sufficient to cover the costs of that investment. But this means the public club has zero net equity (a 100% debt/equity ratio), so it won't work if there is uncertainty.

If new members can be charged a one-time initiation fee, that existing members don't have to pay, this problem can be resolved. That gives existing members an incentive to build equity in the club, so they can issue new shares and sell them to new members at a positive price. But that would mean the club is a Private (excludable) club, and not a Public club.

What happens in a multi-period model, where each period the members enjoy the benefits of previous periods' investment and invest for future periods' enjoyment? The hypothetical social planner would make everyone a club member, and require the club to invest up to the point where the Present Value of the sum (across all future members) of the marginal benefits equals the current marginal cost. A net flow of new members is like investing in an appreciating asset from the Planner's perspective. But if the existing club members themselves vote on how much to invest, they will ignore the benefits that will accrue to new members who join after the investment has been made, and will discount the future returns if they themselves might leave the club. And they will want to postpone investment until the new members actually arrive and share the costs. The existing members will only solve the Planner's Problem if there are no new members.

Take an extreme case, in continuous time, where a discrete investment has to be paid for by existing members at a point in time, but yields a continuous flow of future benefits. That investment will never be made. The proof is simple. If the investment were made at time t, each individual member would resign at t minus epsilon, and rejoin the club at t plus epsilon. They get nearly all the benefits without paying the costs.

The equity in a Public Club is a Public Good. Everyone wants to wait for someone else to put their equity into the lighthouse.

I'm trying to think of a simple but reasonably general way to set up a model with an interior solution that lets us see what the amount of under-investment depends on. No luck yet.

[This is related to my previous post on Open Borders, but I thought it best to try to think more generally and abstractly about the underlying problem.]

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If new members can be charged a one-time initiation fee, that existing members don't have to pay, this problem can be resolved. That gives existing members an incentive to build equity in the club, so they can issue new shares and sell them to new members at a positive price. But that would mean the club is a Private (excludable) club, and not a Public club.

That is the normal way of doing it, in my experience, but there's a number of points here.

The existing members generally do pay a yearly upkeep fee, typically lower than what new members pay. However, there's a balance between attracting new members and bringing more fees, because most new members will stay a few years and probably contribute to the club along the way. Thus, the club takes a risk on new members, in the belief it will pay back in the longer run.

Clubs still charge per-usage fees (e.g. they charge for beer, food, coffee etc and make a profit on those) so in that sense they operate as a regular vending business. Things like lounges, billiard tables, etc in the club are in fact rivalrous goods in as much as one person sitting on the lounge excludes others from sitting in the same spot. This rivalrous nature only shows up occasionally when the place is crowded, so perhaps in an economic analysis it can be ignored, but I don't think it can be entirely ignored. The club wants to be full enough to have a decent atmosphere, but not crowded to be unpleasant, and that comes back to how attractive to make it for new members.

There's no way for either existing members, or new members to extract the equity out of the club (or at least it's very difficult to do so) other than just use the facilities, and the type of facilities tend to self-limit in as much as there's only so many games of billiards you want to play.

One thing I will note, is that in my area clubs are gradually becoming less popular. People keep in touch by electronic means now, and coordinate their activities without "hanging around" any particular area. A lot of the value created by a club used to be meeting with like-minded people, rather than the physical facilities. However, it's not such a big deal any more.

Tel: can you give me any sort of examples where newly-joined members pay a temporarily higher fee than existing members? What sort of clubs are we talking about? (I have very little experience of clubs.)

The main way existing members can extract equity from a club is to use it but not reinvest to cover depreciation, I think.

If you are talking about other types of goods, that are non-excludable but feel a bit similar to club goods, you could look at the case of "Open Source" software, where there's no membership fee imposed (you can just download it for the cost of Internet bandwidth) and yet there is clearly equity built up in the overall software base, and we have plenty of people investing in expanding that base.

Who's to say if there is "under investment" in the Open Source software movement? How would you measure it? People invest what they want to invest, some make money out of it, others don't. Since no one is forced to get involved, we have to presume the level of investment is exactly the desired amount. At least, a great burden would fall on someone attempting to prove a "market failure" and the need for forcible interference.

The question of whether it is "non-rival" is quite difficult. As a regular Linux desktop user, I really don't care whether you do or don't use the same desktop as me. It really doesn't effect me. From that sense it is "non-rival".

However, as a professional programmer, I want enough people to use the same software base that I'm experienced in (thus ensuring demand for those skills) but I don't want everyone to compete with me for jobs (i.e. I want a limited supply of professional skills). You can see this with computer languages... for a long time people were happy with the "perl" language, but then a small bunch started saying "Oh we don't use perl any more, we are all into python now" and the advantage of that is partly the whole "new new thing" mantra that attracts clueless investment dollars, but also the people with those new skills can be the big fish in a small pool. Then a few years later, "Oh we don't use python any more, it's all ruby now" and the process repeats with yet another small pond.

With these break-aways into new areas, there's actually incentive to be the first mover, because you can expect future growth in your particular niche will give a greater benefit to the people who are already in there. That said, not every niche grows, but that's a risk issue which probably should be dealt with separately.

Here's one example, where new members are charged about 50% higher than renewals.

http://brontesurfclub.com.au/new-members/open-membership.html

However, now that you sent me searching, I do see a lot of clubs around my area with quite low yearly membership ($20 per year or lower) and those charge same for everyone. Mind you, I don't think those examples with very cheap membership are really pulling their main revenue from memberships at all. I would guess they make it mostly out of drinks and food, and maybe the poker machines, keno (i.e. gambling).

There's a similar and related problem when generation of something which is basically a public good requires a large lump of capital and returns no value until the total amount of capital is acquired... at which point it returns a windfall value and becomes pretty much a public good.

The response to this has generally been Kickstarter's model. First people *subscribe* to fund whatever-it-is. When enough capital has been *subscribed*, only then do people actually *pay* the capital (avoiding the "I spent money and got nothing" problem), and then the public good is built.

The people who subscribed get... mostly bragging rights, really; early access, thank you notes, and other very minor excludable club goods.

If you think about it this is actually a very hard coordination problem. The solution was developed to fund things like bridges back in the 18th century or earlier.

First, on a pure economic arguments, how risk premia are priced is the whole business in investing under uncertainty – especially if there is an reversibility component. In many cases investments don't get made on the margin by simple net present value calculations because the implied American style option in the investment has so much value. Unless you include these effects, there is little point talking about optimal investment in a multi-period model with uncertainty. God help us all if the social planner thinks only about NPV!

Of course what you are trying to argue is something akin to why we should restrict immigration. But the same problem goes for having children. I got to inherit a world with computers, airplanes, and MRI machines. The fact that I did nothing to help with those investments, and the fact that humanity will benefit forever from those ideas, that did not stop the investments from being made. In your world, since the unborn will benefit from our progress in ways we cannot comprehend, that should kill most research and development. Immigrants and babies who grow up are the same thing.

Nathanael: "The response to this has generally been Kickstarter's model. First people *subscribe* to fund whatever-it-is. When enough capital has been *subscribed*, only then do people actually *pay* the capital (avoiding the "I spent money and got nothing" problem), and then the public good is built."

Interesting. Yes, that does seem to me to be closely related. (I think Frances is planning a post on Kickstarter, but I hadn't seen that connection.)

Avon: there are two (or more) "projects": 1. NPV Invest today 2. max {NPV invest tomorrow, 0}. If new members will arrive tomorrow, and share some of the costs, that raises the NPV to an existing member and raises the option value of waiting for the existing member. It's NPVs all the way down.

"Immigrants and babies who grow up are the same thing."

Not to me, if they are my babies. People are funny like that. So are all mammals.

"Not to me, if they are my babies. People are funny like that. So are all mammals." Some mammals, I suppose - but immigrants are mammals too.

Before you take about how NPV vs option valuation works, read Dixit and Pindyck, Investment Under Uncertainty. That's top notch economic thinking.

Sorry, but you have a narrow welfare redistributionalist view of economics. These "club investments" you keep talking about are not competitive market made decisions - these are political decisions, imposed by government. You are right that open borders mean the end of the welfare state, and I will be happy when it comes. Three hundred years from now we will look at welfare states and their silly borders as ridiculous as the notion of the Divine Right of Kings and the waste in the opulence of Versaille.

This seems to have political implications. If some people think of the state as a private club, it ought to exist for them and their demographic group ["Don't let the Government touch my Medicare], but there is a danger, or demographic certainty, that outsiders will join the club in the future, that group will underinvest in club goods.

If public clubs are leagues of private clubs, children of extant clubs are already factored in. It's the gain or loss of clubs (either through substantial change in proportion or through actual entry/exit) that affects investment. Maybe looking at sports leagues and conferences would be productive?

Great post on a topic I'm trying to understand better. I don't have anything particularly relevant to add, except that I recently read "Bowling Alone" and thought that it was pretty good and had some fairly simple explanations of social capital for a non-economist like myself. I also like Terence Kealey's stuff on science as a collegiate good, in which you have to participate to consume the good but participation is (in principle) open, and how he relates it to the data that government spending on science seems to crowd-out rather than expand scientific research.

In period 1 the utility of an existing member who decides to stay in the club is ln(y-t)+a, where y is their income, t is the tax rate levied by the club to fund the public good in period 2 and a is some other benefit of being in the club in period 1. In period two the benefit of being in a club is ln(N*t) where N is the number of members in period one (constant returns to scale in producing the public good). B is discount factor. So the total lifetime utility of an existing member who decides to stay in the club in first period is

U(stay)=ln(y-t)+a+B*ln(N)+B*ln(t)

If you leave the club in period 1 you don't pay the tax. Assuming you can still "jump in" in period 2 your utility is

U(leave)=ln(y)+B*ln(N)+B*ln(t)

If a is less than ln(1+B) nobody stays in the club in period 1. Yes. BUT, this is regardless of whether there are some outside members (who have no option to be in the club in period 1) who could join in 2.

If a is greater than ln(1+B) then everyone stays in the club in period 1, REGARDLESS of whether some outsiders will join in period 2.

This is just the usual, classic, standard, public goods, free rider problem, which doesn't have anything to do with open borders.

Yes, but that doesn't change anything, or at least it doesn't in Nick's model. Nick's explicitly talking about a non-rival good (clubs/public goods). So who cares if some immigrants come in and enjoy the benefits of institutions that have been build up over generations? This doesn't change the calculus for the so-called "natives".

notsneaky: I'm afraid I didn't quite follow your set-up. This is one example of roughly what I had in mind:

Let the club have a stock of capital K, which provides a benefit to each member a=F(K). And let the cost of investment I be shared among the members N, so each member pays I/N in dues. (Can't use your "t", because we need t for time.) Then standard dK/dt=I-dK. That would be the simplest setup, because it's exactly like the neoclassical, except K produces non-rival goods.

With no entry or exit from the club, so N is constant over time, the existing members will choose I optimally.

There are now two questions:

1. If N is growing exogenously over time, and the existing members N(t) at time t choose I(t), will they choose I(t) optimally? I say they will under-invest, or have an incentive to postpone investment, until the new members arrive and share the costs.

2. If we make membership endogenous, will the timing of joining the club be optimal? I say that potential new members will have an incentive to postpone joining a club where K is growing over time, so they can enjoy the benefits without sharing the costs.

(In this setup, of course, the social planner would force everybody to join the club at the beginning of time.)

>> Who's to say if there is "under investment" in the Open Source software movement? How would you measure it? People invest what they want to invest, some make money out of it, others don't. Since no one is forced to get involved, we have to presume the level of investment is exactly the desired amount. At least, a great burden would fall on someone attempting to prove a "market failure" and the need for forcible interference.

There's clearly underinvestment in open source software. Basic tasks such as documentation are neglected for a significant fraction of open source programs, because strictly voluntary contributions tend to "scratch one's own itch" in terms of feature expansion or bugfixes. Contributors tend to be experts in the program that benefit less from documentation, hence the total value to prospective new members is absent from the calculation. It's almost exactly Nick's proposed dilemma. (Large open-source programs tend to resolve this via paid staff, such as the Mozilla foundation, or by being so significant [as in the Linux kernel] that large organizational contributors see something closer to a "whole ecosystem" benefit rather than an individual's benefit and can pay their own staff for general development work.)

Your subsequent analysis features a network effect, where the value to existing members is an increasing function of the club size. That's an interesting problem, but it's separate from the minimalist "public club" model.

@notsneaky:

>> So who cares if some immigrants come in and enjoy the benefits of institutions that have been build up over generations? This doesn't change the calculus for the so-called "natives".

It does.

Imagine membership in a club was ongoing an irrevocable, and the club membership votes on investment. Investments here cost a fixed amount, but provide per-capita benefits (say, a $1k investment provides $1/period benefit to each member).

If the club members knew with certainty that in the next period the membership would double, they would rationally vote to defer investment to that period, in order to divide the cost over a much larger membership with almost the same flow of benefits (only delayed by one period).

In general, if club membership is fixed at P, all members discount future benefits at rate d, and a unit of investment provides a fixed per-capita benefit of b in each period, then current members will vote to make any investment with gross cost (C) of (b*P)/(d) or less.

If instead club membership is growing at rate r (with current-period membership of P), then members have an additional option to plan to defer the investment to the next period. The choices are then to pay per-capita cost of (c) to receive b/d per-capita benefits (present value), to pay (c/(1+r)) to receive b*(1-d)/(d), or to decline to make the investment.

If (1+r)*(1-d) > 1 (approximately r > d), then deferring investment to the future seems like the wisest choice for any net-positive-value investment. Of course, in the next period current and recently-added members will face the same decision and again the rational choice is to defer the investment.

Under fixed expectations, agents will be continually surprised that the investment isn't made. Under rational expectations this looks like an unstable "omega point" in the indefinite future leading to undetermined behaviour. In reality, the growth rate itself is variable depending in part on the current stock of capital, agents aren't infinitely-lived, and utility is nonlinear, so we'll see mixed results. The common solution is to make at least some of the investment anyway yet grumble about come-lately "freeloaders."

Nick, first the two period model. That's pretty much what I have above where the "t" is the investment. And F(k)=sum of individual t's = N*t (one unit of "taxes" translates into one unit of the public good - this doesn't matter). And in this set up what happens to N in the second period has no effect on choice of investment. Yes, some (maybe all) "natives" may choose not to be part of the club in the first period. But that's because this is a standard public goods problem. If you let people consume public goods without the ability to charge'em for it, then yeah, there'll be under investment. But this happens with or without immigration.

Majormax, I got to work this out but a quick objection to drawing hasty conclusions from a multi-period set up is that yes, it may be that "natives" postpone their investment for reasons you mention, but hey, at the same time, they're getting contributions to the public good fund from natives which they otherwise wouldn't. It's a bit like saying "if you're going to inherit 10 million dollars from your uncle, you'll invest less in the stock market today, so... you're worse off for getting the 10 million dollars!"

I also think there's some false equivocation going on above between "investment per person in the public good will be lower" and "there will be sub-optimal investment". Yes, with population growing, investment (per person) might be lower because present day "natives" are going to try to free-ride on the future contributions of future "migrants" (whose utility they don't care about). But that does not mean that the natives' utility will be lower with immigration than without. They get the future migrants to pay for the public good and enjoy more private consumption themselves.

As an addendum, I forgot to discount the cost to the subsequent period. The "planned payment" should have current-period present value of c*(1-d)/(1+r), with present-value benefits of b*(1-d)/d, leading to the one-period delays cancelling. This means a profit-maximizing, infinitely-lived agent should seek to defer positive-value investment when:

... if I've done the math correctly. Unlike the previous in-error calculation, this will result in investment deferral even if r < d. However, investments are never delayed indefinitely in this model (even with rational expectations), since as the club size becomes infinite the profit of an investment tends towards b/d (per capita cost goes to 0), and r(1-d)/(r+d) < 1.

notsneaky: with the Hotel California assumption, true. But you get the first best investment if existing members can charge new members an entry fee representing a share of the club's equity. (You would need price discrimination if potential new members value that equity differently, and the first best, with perfect price discrimination, would be for existing members to charge new members 99% of their net gains from joining the club.)

> I also think there's some false equivocation going on above between "investment per person in the public good will be lower" and "there will be sub-optimal investment". Yes, with population growing, investment (per person) might be lower because present day "natives" are going to try to free-ride on the future contributions of future "migrants" (whose utility they don't care about). But that does not mean that the natives' utility will be lower with immigration than without. They get the future migrants to pay for the public good and enjoy more private consumption themselves.

Ignoring the attribution of who's welfare is more important (incumbents or new members), this math does suggest that all other things being equal, lower population growth rates lead to more capital-intensive production, and thus lower return on the marginal capital investment.

" But you get the first best investment if existing members can charge new members an entry fee representing a share of the club's equity."

At least in the two period model I'm not seeing this. The choice of investment is independent of immigration in second period. Maybe if you put in some serious indivisibilities or something...

And "first best" for whom? I mean, yes, if the existing members can charge the new members a fee, they will be better off. But same would be true if they could levy a "tax on foreigners, living abroad", per Monty Python. There's no reason to think that they would be worse off with immigration-but-no-fee than no-immigration.

Majromax, yes I know what a non-rival good is. But the way it usually works is that you pay a fixed cost F first, then you get benefits, b, later. These benefits b are independent of size of fixed cost F, that's what makes it a fixed cost rather than variable cost. This is true whether a good is rival or not. Maybe I'm missing something...

"Even if there are positive net benefits for some members in the first period (because the club has some other benefits), the members will under-invest in the first period if they cannot charge new members an additional premium in the second period."

the period should come after the words "first period" and the last clause beginning with "if they..." should be omitted because it doesn't follow.

notsneaky: 2 period model. 0% discount rate. N(1) members first period, N(2) members second period. The club invests K in the first period, at a cost U= -K/N(1) per member, and each member gets benefits U=log(K) in the second period.

The N(1) members choose K to maximise [log(K) - K/N(1)].

If they could borrow to finance K, so all N(2) members share the cost, they would choose K to maximise [log(K) - K/N(2)].

And a utilitarian social planner would maximise [N(2).log(K) - N(1).K/N(1)]

1. What I was comparing was the situation "with no migration" to the situation "with migration but no borrowing" (I got confused for a second there above). Let's say there's only one native (avoids issues of free riding among the natives themselves) so N1=1. In the first situation the utility function is log(K)-K so K=1. In the second situation the utility function is... log(K)-K. So still 1. So migration by itself will not lower investment in the public good.

2. Ok, but let's compare "migration but no borrowing" to "migration with borrowing". Yes, in that case K=N2>1 and the native is better off. But. Is she better off because of the ability to borrow or simply because now part of the cost is paid by the immigrants? Suppose there's no migration but the ability to borrow. With no discounting the utility function is ... log(K)-K. So borrowing does not make a difference. Hence, it's the ability to charge the immigrants a fee that makes the native better off. I could just as well set up the utility function as log(K)-K+T, where T is the entry tax collected by the native from the migrants and claim that the native is better off.

Yes, if you can get someone else to pay for what you want to consume you will a) be better off and b) likely consume more of it. But this has nothing to do with public goods or clubs. Everything here could be private, rivalrous, excludable. And in that case I don't see any reason to bring public goods or clubs into it, just go all the way and postulate that the native will maximize their utility if they charge the immigrants their maximum willingness to pay.

3. Can the ability to charge migrants/borrow ameliorate the free riding problem? Not necessarily. Suppose there's a small benefit to being in the club in period 1, given by a. And natives (now more than 1) can jump in and out of the club.

With no immigration the utility of a native who stays in the club is ln(K)-K+a. The utility of native who leaves the club then jumps back in is ln(K). With optimal K=1, the natives will only undertake investment if a>1.

With immigration but no borrowing it's... exactly the same. Utility functions are the same, the condition a>1 is the same.

With immigration and borrowing the utility function of a native who stays in period 1 is ln(K)-K/N2+a (where N2>1). Utility of a native who free rides is ln(K). Now optimal K=N2. So utility of those who stay is ln(N2)-1+a. Utility of those who free ride is ln(N2). So even though the utility functions are different the condition a>1 is still the same. So the ability to borrow and charge migrants doesn't even do anything to fix the free riding problem if it exists.

I can sort of see how there might be some model where the immigration/borrowing thing serves as some kind of amplification mechanism to an existing problem, but then one can also think of one where it works the opposite.

The amount T the natives can charge the immigrants (the immigrant's willingness to pay) is an increasing function of K. In fact, T=log(K) per immigrant. Which gives the natives the incentive to invest more.

Sure, assuming that the only reason for migration is access to the public good in the host country and that there is no public good in the home country. This is the "if you can get someone else to pay for your consumption you'll consume more" effect.

Hence, it's the ability to charge the immigrants a fee that makes the native better off. I could just as well set up the utility function as log(K)-K+T, where T is the entry tax collected by the native from the migrants and claim that the native is better off.

Yes, you could say that. If immigrants are willing to pay this entry fee, it demonstrates that there is value to be had (from the immigrant perspective).

Thus, if the natives give value away for free they are making an economic loss (leaving their money on the table I think is the expression). Just like any shop keeper would attempt to profit by selling goods.

Sure, assuming that the only reason for migration is access to the public good in the host country and that there is no public good in the home country. This is the "if you can get someone else to pay for your consumption you'll consume more" effect.

In any transaction, both sides would like the other side to pay for their additional consumption. I would like free goods in every shop, free housing, and I would also like plenty of free time.

There's no special "effect" here, it's just the price setting nature of transactions. How else would you suggest we do it otherwise?

There's clearly underinvestment in open source software. Basic tasks such as documentation are neglected for a significant fraction of open source programs, because strictly voluntary contributions tend to "scratch one's own itch" in terms of feature expansion or bugfixes. Contributors tend to be experts in the program that benefit less from documentation, hence the total value to prospective new members is absent from the calculation. It's almost exactly Nick's proposed dilemma.

What I hear you saying is that the level of investment is about optimal to satisfy the needs of those who contributed significant investment effort. I still cannot see a problem here, seems like exactly what you would expect.

At least, there's no evidence anyone is being harmed by this state of affairs.

From my own perspective, I must say I find all the necessary information very easy to obtain. Most common tasks can be understood quite quickly; just use a web search and someone will have precise step by step instructions.

Far better overall than my experience with manufacturers proprietary documentation. I've wasted plenty of time chasing down obscure and meaningless error messages from supposedly "enterprise" HP and/or IBM gear, and it still leaves all concerned scratching their heads. Reading source code is actually faster than hanging on the line to a support department.